Preparations

Load the necessary libraries

library(gbm)         #for gradient boosted models
library(car)
library(dismo)
library(pdp)
library(ggfortify)
library(randomForest)
library(tidyverse)
library(gridExtra)
library(patchwork)

Scenario

Verneaux (1973) measured the abundance of 27 fish species and 11 environmental variables from 30 locations along the Doubs river which runs along the France-Switzerland border. The environmental data comprised a range of hydrology, geomorphology and chemistry parameters and amungst other things, the ichthyologist was interested in relating fish abundances to the environmental drivers.

The data are in two files.

verneaux.fish.csv - the abundance of 27 fish species
verneaux.env.csv - the environmental data

Variable	Description
DAS	Distance from source (km)
ALT	Altitude (m above sea level)
PEN	Slope (per thousand)
DEB	Mean minimum dischange (m^3.s-1)
PH	Water pH
DUR	Calcium concentration (hardness) (mgL^-1)
PHO	Phosphorus concentration (mgL^-1)
NIT	Nitrate concentration (mgL^-1)
AMM	Ammonium concentration (mgL^-1)
OXY	Dissolved oxygen (mgL^-1)
BDO	Biological oxygen demand (mgL^-1)

Read in the data

fish = read_csv('../public/data/verneaux.fish.csv', trim_ws=TRUE)
glimpse(fish)

## Rows: 30
## Columns: 27
## $ CHA <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 2, 1, 1, 0, 0, 0, 0, …
## $ TRU <dbl> 3, 5, 5, 4, 2, 3, 5, 0, 0, 1, 3, 5, 5, 5, 4, 3, 2, 1, 0, 0, 0, 0, …
## $ VAI <dbl> 0, 4, 5, 5, 3, 4, 4, 0, 1, 4, 4, 4, 5, 5, 4, 3, 4, 3, 3, 1, 1, 0, …
## $ LOC <dbl> 0, 3, 5, 5, 2, 5, 5, 0, 3, 4, 1, 4, 2, 4, 5, 5, 4, 3, 5, 2, 1, 1, …
## $ OMB <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 2, 0, 1, 1, 0, 0, 0, 0, …
## $ BLA <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 2, 1, 1, 0, 0, 0, …
## $ HOT <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 3, …
## $ TOX <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 3, 3, 2, 2, 2, …
## $ VAN <dbl> 0, 0, 0, 0, 5, 1, 1, 0, 0, 2, 0, 0, 0, 0, 3, 5, 3, 2, 2, 2, 2, 3, …
## $ CHE <dbl> 0, 0, 0, 1, 2, 2, 1, 0, 5, 2, 1, 1, 0, 1, 3, 2, 2, 3, 1, 3, 2, 4, …
## $ BAR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 2, 4, 4, 5, …
## $ SPI <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 2, 3, 2, 1, …
## $ GOU <dbl> 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 1, 2, 4, 4, 5, 5, …
## $ BRO <dbl> 0, 0, 1, 2, 4, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 3, 3, …
## $ PER <dbl> 0, 0, 0, 2, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 1, 2, 3, 4, …
## $ BOU <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 3, …
## $ PSO <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, …
## $ ROT <dbl> 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, …
## $ CAR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 3, …
## $ TAN <dbl> 0, 0, 0, 1, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 4, 4, 4, …
## $ BCO <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, …
## $ PCH <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, …
## $ GRE <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, …
## $ GAR <dbl> 0, 0, 0, 0, 5, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 1, 2, 2, 5, 5, 5, 5, …
## $ BBO <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, …
## $ ABL <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 5, 5, …
## $ ANG <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, …

env = read_csv('../public/data/verneaux.env.csv', trim_ws=TRUE)
glimpse(env)

## Rows: 30
## Columns: 11
## $ DAS <dbl> 0.3, 2.2, 10.2, 18.5, 21.5, 32.4, 36.8, 49.1, 70.5, 99.0, 123.4, 1…
## $ ALT <dbl> 934, 932, 914, 854, 849, 846, 841, 792, 752, 617, 483, 477, 450, 4…
## $ PEN <dbl> 48.0, 3.0, 3.7, 3.2, 2.3, 3.2, 6.6, 2.5, 1.2, 9.9, 4.1, 1.6, 2.1, …
## $ DEB <dbl> 0.84, 1.00, 1.80, 2.53, 2.64, 2.86, 4.00, 1.30, 4.80, 10.00, 19.90…
## $ PH  <dbl> 7.9, 8.0, 8.3, 8.0, 8.1, 7.9, 8.1, 8.1, 8.0, 7.7, 8.1, 7.9, 8.1, 8…
## $ DUR <dbl> 45, 40, 52, 72, 84, 60, 88, 94, 90, 82, 96, 86, 98, 98, 86, 88, 92…
## $ PHO <dbl> 0.01, 0.02, 0.05, 0.10, 0.38, 0.20, 0.07, 0.20, 0.30, 0.06, 0.30, …
## $ NIT <dbl> 0.20, 0.20, 0.22, 0.21, 0.52, 0.15, 0.15, 0.41, 0.82, 0.75, 1.60, …
## $ AMM <dbl> 0.00, 0.10, 0.05, 0.00, 0.20, 0.00, 0.00, 0.12, 0.12, 0.01, 0.00, …
## $ OXY <dbl> 12.2, 10.3, 10.5, 11.0, 8.0, 10.2, 11.1, 7.0, 7.2, 10.0, 11.5, 12.…
## $ DBO <dbl> 2.7, 1.9, 3.5, 1.3, 6.2, 5.3, 2.2, 8.1, 5.2, 4.3, 2.7, 3.0, 2.4, 3…

Data preparation

For this example, we are going to focus on total fish abundance and therefore we need to sum the total number of fish within each location (rows).

fish <- fish %>% rowSums

Exploratory data analysis

car::scatterplotMatrix(cbind(fish, env))

Fit the model

Now we will specify the gradient boosted model with:

Poisson distribution. This is not the same as the usual ‘family’. It does not directly nominate a modelling family. Instead it determines the loss function that is used to evaluate the tree splits.
var.monotone: indicates which of the predictors should be constrained to monotonic trends (0: not monotonic, 1: positive monotonic, -1: negative monotonic)
n.trees: the total number of trees to sequentially fit
interaction.depth: the number of splits within each tree. This roughly equates to the order of interactions. The higher the number, the more complex the interactions can be, however, the faster the potential rate of learning
bag.fraction: random fraction of data used to fit a single tree
shrinkage: rate of learning. The smaller the value, the slower the learning rate. Typically, values should be between 0.001 and 0.01.
train.fraction: the proportion of the data used to train the tree. This is for the purpose of determining the balance between over and underfitting and evaluate the accuracy of the fit, however for the former purpose, this is a cruder approach than either out-of-bag or cross validation. Therefore, it is arguably better to set this fraction to 1 (use 100 percent of the data for training) and use either out-of-bag or cross validation and (as we have done here), withhold a fraction of the full data and perform our own accuracy checking.
cv.folds: the number of cross validation folds.

The values of n.trees, interaction.depth and shrinkage used below are purely based on what are typically good starting points. Nevertheless, they will be data set dependent. We will start off with those values and then evaluate whether they are appropriate.

fish.gbm = gbm(fish ~ DAS + ALT + PEN + DEB + PH + DUR + PHO + NIT + AMM +
              OXY + DBO,
              data=env,
              var.monotone = c(1,-1,-1,1,0,1,0,0,0,0,0),
              distribution='poisson',
              n.trees=10000,
              n.minobsinnode = 2,
              interaction.depth=5,
              bag.fraction=0.5,
              shrinkage=0.01,
              train.fraction=1,
              cv.folds=3)

We will now determine the optimum number of trees estimated to be required in order to achieve a balance between bias (biased towards the exact observations) and precision (variability in estimates). Ideally, the optimum number of trees should be close to 1000. If it is much less (as in this case), it could imply that the tree learned too quickly. On the other hand, if the optimum number of trees is very close to the total number of fitted trees, then it suggests that the optimum may not actually have occured yet and that more trees should be used (or a faster learning rate).

(best.iter = gbm.perf(fish.gbm,method='OOB'))

## [1] 262
## attr(,"smoother")
## Call:
## loess(formula = object$oobag.improve ~ x, enp.target = min(max(4, 
##     length(x)/10), 50))
## 
## Number of Observations: 10000 
## Equivalent Number of Parameters: 39.99 
## Residual Standard Error: 0.003112

(best.iter = gbm.perf(fish.gbm,method='cv'))

## [1] 525

Conclusions:

both out-of-bag and cross validation suggest that the total number of trees greatly exceeds the number necessary to achieve the optimum balance between bias and precision.
it might be worth decreasing the learning rate.
we can probably also reduce the total number of trees (so that it runs a little quicker)

fish.gbm = gbm(fish ~ DAS + ALT + PEN + DEB + PH + DUR + PHO + NIT + AMM +
              OXY + DBO,
              data=env,
              var.monotone = c(1,-1,-1,1,0,1,0,0,0,0,0),
              distribution='poisson',
              n.trees=10000,
              n.minobsinnode = 2,
              interaction.depth=5,
              bag.fraction=0.5,
              shrinkage=0.001,
              train.fraction=1,
              cv.folds=3)

(best.iter = gbm.perf(fish.gbm,method='OOB'))

## [1] 2589
## attr(,"smoother")
## Call:
## loess(formula = object$oobag.improve ~ x, enp.target = min(max(4, 
##     length(x)/10), 50))
## 
## Number of Observations: 10000 
## Equivalent Number of Parameters: 39.99 
## Residual Standard Error: 0.0008918

(best.iter = gbm.perf(fish.gbm,method='cv'))

## [1] 3008

Conclusions:

both out-of-bag and cross validation suggest that the total number of trees greatly exceeds the number necessary to achieve the optimum balance between bias and precision.
it might be worth decreasing the learning rate.

Explore relative influence

If a predictor is an important driver of the patterns in the response, then many of the tree splits should feature this predictor. It thus follows that the number of proportion of total splits that features each predictor will be a measure of the relative influence of each of the predictors.

summary(fish.gbm, n.trees=best.iter)

Conclusions:

shell weight is overwelmingly the most influential predictor.
we can also ascribe a kind of signficance to the influence values by determining which of them are greater than \(100/p\) where \(p\) is the number of predictors. The logic here is that if all predictors were equally influential, then they would all have a relative influence of \(100/p\). Hence, any predictors that have a relative influence greater than this number must be explaining more than its share (and more than pure chance) and predictors with relative influence lower are explaining less than chance (and therefore not significantly influential).

Explore partial effects

attr(fish.gbm$Terms,"term.labels")

##  [1] "DAS" "ALT" "PEN" "DEB" "PH"  "DUR" "PHO" "NIT" "AMM" "OXY" "DBO"

plot(fish.gbm, 1, n.tree=best.iter)

plot(fish.gbm, 2, n.tree=best.iter)

fish.gbm %>%
    pdp::partial(pred.var='DAS',
                 n.trees=best.iter,
                 recursive=FALSE,
                 inv.link=exp) %>%
    autoplot()

Recursive indicates that a weighted tree traversal method described by Friedman 2001 (which is very fast) should be used (only works for gbm). Otherwise a slower brute force method is used. If want to back transform - need to use brute force.

nms <- attr(fish.gbm$Terms,"term.labels")
p <- vector('list', length(nms))
names(p) <- nms
for (nm in nms) {
  print(nm)
  p[[nm]] <- fish.gbm %>% pdp::partial(pred.var=nm,
                                 n.trees=best.iter,
                                 inv.link=exp,
                                 recursive=FALSE,
                                 type='regression') %>%
      autoplot() +
      ylim(18, 60)
}

## [1] "DAS"
## [1] "ALT"
## [1] "PEN"
## [1] "DEB"
## [1] "PH"
## [1] "DUR"
## [1] "PHO"
## [1] "NIT"
## [1] "AMM"
## [1] "OXY"
## [1] "DBO"

patchwork::wrap_plots(p)

#do.call('grid.arrange', p)

We might also want to explore interactions…

fish.gbm %>%
    pdp::partial(pred.var=c('DAS'),
                 n.trees=best.iter, recursive=FALSE, inv.link=exp) %>%
    autoplot()

fish.gbm %>%
    pdp::partial(pred.var=c('DAS','ALT'),
                 n.trees=best.iter, recursive=TRUE) %>%
    autoplot()

g1 = fish.gbm %>% pdp::partial(pred.var='DAS', n.trees=best.iter,
                            recursive=FALSE,inv.link=exp) %>%
    autoplot
g2 = fish.gbm %>% pdp::partial(pred.var='ALT', n.trees=best.iter,
                            recursive=FALSE,inv.link=exp) %>%
    autoplot


g1 + g2

Explore accuracy

fish.acc <- env %>%
    bind_cols(Pred = predict(fish.gbm,
                             newdata=env,
                             n.tree=best.iter,
                             type='response'))

with(fish.acc,  cor(fish, Pred))

## [1] 0.9905805

fish.acc %>%
  ggplot() +
  geom_point(aes(y=Pred,  x=fish))

Explore interactions

Computes Friedman’s H-statistic to assess the strength of variable interactions. This measures the relative strength of interactions in models It is on a scale of 0-1, where 1 is very strong interaction In y=β_0+β_1x_1+β_2x_2+β_3x_3.. H= If both main effects are weak, then the H- stat will be unstable.. and could indicate a strong interaction.

One at a time

What were the strong main effects: - DAS - ALT - OXY

attr(fish.gbm$Terms,"term.labels")

##  [1] "DAS" "ALT" "PEN" "DEB" "PH"  "DUR" "PHO" "NIT" "AMM" "OXY" "DBO"

interact.gbm(fish.gbm, env,c(1,2), n.tree=best.iter)

## [1] 0.03754515

interact.gbm(fish.gbm, env,c(1,10), n.tree=best.iter)

## [1] 0.1102318

interact.gbm(fish.gbm, env,c(2,10), n.tree=best.iter)

## [1] 0.05219991

interact.gbm(fish.gbm, env,c(1,2,10), n.tree=best.iter)

## [1] 0.003372834

fish.gbm %>% pdp::partial(pred.var=c('DAS', 'OXY'),  n.trees=best.iter, recursive=FALSE) %>%
    autoplot

fish.gbm %>% pdp::partial(pred.var=c('DAS', 'OXY'),
                          n.trees=best.iter,
                          recursive=FALSE,
                          inv.link = exp) %>%
    autoplot

fish.gbm %>% pdp::partial(pred.var=c(1, 10),  n.trees=best.iter, recursive=FALSE) %>% autoplot

fish.gbm %>% pdp::partial(pred.var=c(2, 10),  n.trees=best.iter, recursive=FALSE) %>% autoplot

fish.grid = plot(fish.gbm, c(1,10), n.tree=best.iter, return.grid=TRUE)
head(fish.grid)

ggplot(fish.grid, aes(y=DAS, x=OXY)) +
    geom_tile(aes(fill=y)) +
    geom_contour(aes(z=y)) +
    scale_fill_gradientn(colors=heat.colors(10))

All two way interactions

## [1] "i= 1  Name =  DAS"
## [1] "j= 2  Name =  ALT"
## [1] "i= 1  Name =  DAS"
## [1] "j= 3  Name =  PEN"
## [1] "i= 1  Name =  DAS"
## [1] "j= 4  Name =  DEB"
## [1] "i= 1  Name =  DAS"
## [1] "j= 5  Name =  PH"
## [1] "i= 1  Name =  DAS"
## [1] "j= 6  Name =  DUR"
## [1] "i= 1  Name =  DAS"
## [1] "j= 7  Name =  PHO"
## [1] "i= 1  Name =  DAS"
## [1] "j= 8  Name =  NIT"
## [1] "i= 1  Name =  DAS"
## [1] "j= 9  Name =  AMM"
## [1] "i= 1  Name =  DAS"
## [1] "j= 10  Name =  OXY"
## [1] "i= 1  Name =  DAS"
## [1] "j= 11  Name =  DBO"
## [1] "i= 2  Name =  ALT"
## [1] "j= 3  Name =  PEN"
## [1] "i= 2  Name =  ALT"
## [1] "j= 4  Name =  DEB"
## [1] "i= 2  Name =  ALT"
## [1] "j= 5  Name =  PH"
## [1] "i= 2  Name =  ALT"
## [1] "j= 6  Name =  DUR"
## [1] "i= 2  Name =  ALT"
## [1] "j= 7  Name =  PHO"
## [1] "i= 2  Name =  ALT"
## [1] "j= 8  Name =  NIT"
## [1] "i= 2  Name =  ALT"
## [1] "j= 9  Name =  AMM"
## [1] "i= 2  Name =  ALT"
## [1] "j= 10  Name =  OXY"
## [1] "i= 2  Name =  ALT"
## [1] "j= 11  Name =  DBO"
## [1] "i= 3  Name =  PEN"
## [1] "j= 4  Name =  DEB"
## [1] "i= 3  Name =  PEN"
## [1] "j= 5  Name =  PH"
## [1] "i= 3  Name =  PEN"
## [1] "j= 6  Name =  DUR"
## [1] "i= 3  Name =  PEN"
## [1] "j= 7  Name =  PHO"
## [1] "i= 3  Name =  PEN"
## [1] "j= 8  Name =  NIT"
## [1] "i= 3  Name =  PEN"
## [1] "j= 9  Name =  AMM"
## [1] "i= 3  Name =  PEN"
## [1] "j= 10  Name =  OXY"
## [1] "i= 3  Name =  PEN"
## [1] "j= 11  Name =  DBO"
## [1] "i= 4  Name =  DEB"
## [1] "j= 5  Name =  PH"
## [1] "i= 4  Name =  DEB"
## [1] "j= 6  Name =  DUR"
## [1] "i= 4  Name =  DEB"
## [1] "j= 7  Name =  PHO"
## [1] "i= 4  Name =  DEB"
## [1] "j= 8  Name =  NIT"
## [1] "i= 4  Name =  DEB"
## [1] "j= 9  Name =  AMM"
## [1] "i= 4  Name =  DEB"
## [1] "j= 10  Name =  OXY"
## [1] "i= 4  Name =  DEB"
## [1] "j= 11  Name =  DBO"
## [1] "i= 5  Name =  PH"
## [1] "j= 6  Name =  DUR"
## [1] "i= 5  Name =  PH"
## [1] "j= 7  Name =  PHO"
## [1] "i= 5  Name =  PH"
## [1] "j= 8  Name =  NIT"
## [1] "i= 5  Name =  PH"
## [1] "j= 9  Name =  AMM"
## [1] "i= 5  Name =  PH"
## [1] "j= 10  Name =  OXY"
## [1] "i= 5  Name =  PH"
## [1] "j= 11  Name =  DBO"
## [1] "i= 6  Name =  DUR"
## [1] "j= 7  Name =  PHO"
## [1] "i= 6  Name =  DUR"
## [1] "j= 8  Name =  NIT"
## [1] "i= 6  Name =  DUR"
## [1] "j= 9  Name =  AMM"
## [1] "i= 6  Name =  DUR"
## [1] "j= 10  Name =  OXY"
## [1] "i= 6  Name =  DUR"
## [1] "j= 11  Name =  DBO"
## [1] "i= 7  Name =  PHO"
## [1] "j= 8  Name =  NIT"
## [1] "i= 7  Name =  PHO"
## [1] "j= 9  Name =  AMM"
## [1] "i= 7  Name =  PHO"
## [1] "j= 10  Name =  OXY"
## [1] "i= 7  Name =  PHO"
## [1] "j= 11  Name =  DBO"
## [1] "i= 8  Name =  NIT"
## [1] "j= 9  Name =  AMM"
## [1] "i= 8  Name =  NIT"
## [1] "j= 10  Name =  OXY"
## [1] "i= 8  Name =  NIT"
## [1] "j= 11  Name =  DBO"
## [1] "i= 9  Name =  AMM"
## [1] "j= 10  Name =  OXY"
## [1] "i= 9  Name =  AMM"
## [1] "j= 11  Name =  DBO"
## [1] "i= 10  Name =  OXY"
## [1] "j= 11  Name =  DBO"

Tuning

The takes a long time - do over a break

fish.gbm1 <- dismo::gbm.step(data=cbind(fish, env) %>% as.data.frame, gbm.x=2:11, gbm.y=1,
                        tree.complexity=5,
                        learning.rate=0.001,
                      n.minobsinnode = 2,
                        bag.fraction=0.5,
                      n.train = 1,
                        n.trees=10000,
                        family='poisson')

summary(abalone.gbm1)

Random Forest

library(randomForest)
fish.rf = randomForest(fish ~ DAS + ALT + PEN + DEB + PH + DUR + PHO + NIT + AMM +
                           OXY + DBO,
                       data=env, importance=TRUE,
                       ntree=1000)
fish.imp = randomForest::importance(fish.rf)
## Rank by either:
## *MSE (mean decrease in accuracy)
## For each tree, calculate OOB prediction error.
## This also done after permuting predictors.
## Then average diff of prediction errors for each tree
## *NodePurity (mean decrease in node impurity)
## Measure of the total decline of impurity due to each
## predictor averaged over trees
100*fish.imp/sum(fish.imp)

##         %IncMSE IncNodePurity
## DAS  0.10514608     19.578297
## ALT  0.08822959     15.527627
## PEN  0.03681748      7.102072
## DEB  0.08942111     18.600003
## PH  -0.01288507      1.065126
## DUR  0.03792810      8.052133
## PHO  0.05512223      5.574289
## NIT  0.06791615      8.942640
## AMM  0.05865963      5.675618
## OXY  0.04979067      4.982862
## DBO  0.05360180      4.269585

varImpPlot(fish.rf)

## use brute force
fish.rf %>% pdp::partial('DAS') %>% autoplot

fish.rf.acc <- env %>%
    bind_cols(Pred = predict(fish.rf,
                             newdata=env))

with(fish.rf.acc,  cor(fish, Pred))

## [1] 0.9843458

fish.rf.acc %>%
  ggplot() +
  geom_point(aes(y=Pred,  x=fish)) +
  geom_point(data = fish.acc, aes(y=Pred,  x=fish), colour = 'red')

Regression Trees Part3

Murray Logan

24 August, 2022

Preparations

Scenario

Read in the data

Data preparation

Exploratory data analysis

Fit the model

Explore relative influence

Explore partial effects

Explore accuracy

Explore interactions

One at a time

All two way interactions

Tuning

Random Forest